On limiting trace inequalities for vectorial differential operators

On Inequalities for Differential Operators

In this paper we study the following problem: Given that certain functionals of u and its derivatives belong to given Lclasses over the infinite interval, what can be said about the L-classes of other functionals? Utilizing a simple device from the theory of linear differential equations, we obtain a number of results due to Landau, Kolmogoroff, Halperin-von Neumann, and Nagy, together with som...

Trace Formulae for Matrix Integro-Differential Operators

where λ is a spectral parameter, Y (x) = [yk(x)]k=1,d is a column vector, Q(x) and M(x, t) are d×d real symmetric matrix-valued functions, and h and H are d×d real symmetric constant matrices. M(x, t) is an integrable function on the set D0 def ={(x, t) : 0≤ t ≤ x ≤ π, x, t ∈ R}, Q ∈ C1[0,π], where C1[0,π] denotes a set whose element is a continuously differentiable function on [0,π]. In partic...

Weighted Inequalities for Potential Operators on Differential Forms

Full Expansion of the Heat Trace for Cone Differential Operators

The operator e−tA and the heat trace Tr e−tA, for t > 0, are investigated in the case when A is an elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter–ellipticity) we obtain a full asymptotic expansion in t of the heat trace as t → 0+. As in the smooth compact case, the problem is reduced to the investigation of the resolvent (A...

Lieb-thirring Inequalities for Higher Order Differential Operators

We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order γ≥3/4 for a fourth order operator in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of such operators, in dimensions greater than one. For the critical case γ=1−1/(2l) in dimension d=1 with l≥2 we prove the inequality L0l,γ,d < Ll,γ,d , which holds in contrast to cu...